Generalized de Sitter Space in $n$-dimensional Minkowski Space

TL;DR

This paper introduces a generalized model of de Sitter space in n-dimensional Minkowski space by replacing the radius with a function, analyzing its geometric and causal properties, and computing curvature for specific cases.

Contribution

It extends de Sitter space by replacing the radius with a function, providing conditions for causal character and curvature analysis in the generalized model.

Findings

Hypersurfaces can be timelike, null, or spacelike based on the function f.

Warped product geometry characterizes non-null hypersurfaces.

Curvature tensors are computed for specific 4D examples.

Abstract

In this paper, we generalize the defining equation for de Sitter space by replacing the de Sitter radius with a function $f$ satisfying certain conditions; each resulting hypersurface is diffeomorphic to de Sitter space, and has a geometry (and causal character) which is controlled by the choice of $f$. Necessary and sufficient conditions are obtained for a hypersurface to be timelike, null, or spacelike in the generalized model; in the non-null case, the geometry is given by a warped product. Several examples of timelike, null, and spacelike hypersurfaces are presented. Lastly, we calculate the Ricci tensor and scalar curvature for a special family of 4-dimensional generalized de Sitter spaces.